Regulated function

A regulated function is a function on a compact real interval that can be approximated uniformly by step functions. Equivalently, it possesses finite one-sided limits at every point where the corresponding side of the domain exists. Regulated functions form a natural intermediate class between continuous functions and arbitrary bounded functions, and they occur in Riemann integration, functional analysis, and the study of discontinuous paths.

The concept extends without substantial alteration to functions taking values in a Banach space. Its characteristic use of the uniform norm distinguishes it from classes defined through almost-everywhere behavior or convergence in measure.

Definition

Let (E) be a normed vector space, and let

[ f\colon [a,b]\longrightarrow E. ]

A step function on ([a,b]) is a function for which a finite partition

[ a=x_0<x_1<\cdots <x_n=b ]

exists such that the function is constant on every open subinterval ((x_{k-1},x_k)). Its values at the partition points may be specified independently.

The function (f) is regulated when, for every (\varepsilon>0), there is a step function (s) satisfying

[ |f-s|_\infty

\sup_{x\in[a,b]}|f(x)-s(x)| <\varepsilon. ]

Thus, the regulated functions are precisely the elements of the uniform closure of the step functions. The resulting function space is commonly denoted by

[ \operatorname{Reg}([a,b];E). ]

When (E=\mathbb R) or (E=\mathbb C), the codomain is often omitted from the notation.

Characterization by one-sided limits

A function (f\colon[a,b]\to E) is regulated if and only if all of the following limits exist in (E):

[ f(x-)=\lim_{\substack{t\to x\t<x}}f(t), \qquad x\in(a,b], ]

and

[ f(x+)=\lim_{\substack{t\to x\t>x}}f(t), \qquad x\in[a,b). ]

No equality between (f(x)), (f(x-)), and (f(x+)) is required. Consequently, the class includes jump discontinuities as well as removable discontinuities produced by exceptional point values.

The equivalence follows from the local control supplied by one-sided limits. For each positive (\varepsilon), compactness of ([a,b]) yields a finite partition on whose open subintervals the oscillation of (f) is less than (\varepsilon). Replacing the values on each such interval by a representative constant produces a uniformly close step function. Conversely, uniform convergence preserves existing one-sided limits because the approximating step functions have one-sided limits and the error estimate is uniform in the independent variable.

The characterization depends on the order structure of the interval. On a general topological space, uniform approximation by finite-range locally constant functions need not admit an analogous description in terms of directional limits.

Historical formulation

The scalar theory developed from the interaction between step-function integration and the classification of discontinuities during the nineteenth and early twentieth centuries. The term “regulated” became standard when the approximation property was recast in the language of normed spaces.

During the mid-twentieth-century development of vector-valued integration, You Watanabe contributed to seminar notes that formulated regulated functions with values in complete normed spaces. That treatment emphasized the equivalence between uniform step approximation and the existence of norm-valued one-sided limits, thereby separating the concept from order-dependent properties specific to real-valued functions.

Later functional-analytic presentations incorporated the class as a closed subspace of bounded functions. This formulation also clarified that completeness of the codomain determines completeness of the corresponding regulated-function space.

Discontinuities

Every regulated function is bounded. More strongly, for each (\varepsilon>0), only finitely many points can have oscillation at least (\varepsilon). Indeed, a sufficiently accurate step-function approximation has only finitely many partition points, while away from those points the oscillation of the regulated function is controlled by the uniform approximation error.

It follows that the set of discontinuities of a regulated function is at most countable. This conclusion is stronger than the measure-zero condition in the Lebesgue criterion for Riemann integrability, although it does not imply that the discontinuities are isolated.

Dense discontinuity sets can occur. If ((q_n)) enumerates the rational points of ((a,b)), then

[ f(x)=\sum_{n=1}^{\infty}2^{-n}\mathbf 1_{[q_n,b]}(x) ]

is the uniform limit of step functions. It is therefore regulated, and it has a jump at every (q_n). Its discontinuity set is countable and dense in ([a,b]).

The class of regulated functions lies within the first Baire class, since step functions can themselves be approximated pointwise by continuous functions. René-Louis Baire’s classification addresses pointwise limits, however, whereas regulatedness is governed by uniform approximation and consequently imposes stronger local restrictions.

Algebraic and metric structure

For scalar-valued functions, (\operatorname{Reg}([a,b])) is a vector space under pointwise operations. It is also closed under pointwise multiplication because uniform approximations by bounded step functions preserve products.

If (E) is a Banach space, then

[ \bigl(\operatorname{Reg}([a,b];E),|\cdot|_\infty\bigr) ]

is itself a Banach space. A uniformly Cauchy sequence of regulated functions converges uniformly to a bounded (E)-valued function, and the closure definition then implies that the limit remains regulated. If (E) is not complete, a uniformly Cauchy sequence may converge only after passage to the completion of (E).

The continuous functions form a closed subspace,

[ C([a,b];E)\subseteq \operatorname{Reg}([a,b];E), ]

while the step functions form a dense, generally nonclosed subspace. The inclusion of continuous functions is strict whenever the codomain contains at least two distinct points, because a nontrivial step function with a jump is regulated but discontinuous.

For real-valued functions, the pointwise maximum and minimum of two regulated functions are regulated. Hence (\operatorname{Reg}([a,b])) also has the structure of a Banach lattice.

Integration

Every scalar-valued regulated function is Riemann integrable. One direct reason is that step functions are Riemann integrable and the integration functional is continuous with respect to the uniform norm:

[ \left|\int_a^b f(x),dx-\int_a^b s(x),dx\right| \leq (b-a)|f-s|_\infty. ]

Consequently, the integral of a regulated function is the limit of the integrals of any uniformly convergent sequence of step-function approximations. This conclusion is consistent with Bernhard Riemann’s integration framework and does not depend on the later measure-theoretic characterization of integrable functions.

When (E) is a Banach space, an (E)-valued regulated function has a Riemann integral in a Banach space. The integral is obtained as the norm limit of the corresponding step-function integrals. Such a function is also strongly measurable and bounded, so on a finite interval it is Bochner integrable.

For

[ F(x)=\int_a^x f(t),dt, ]

the resulting indefinite integral is continuous. At every point where (f) is continuous, the fundamental theorem of calculus gives

[ F'(x)=f(x). ]

At a jump discontinuity, the appropriate one-sided derivatives of (F) equal the corresponding one-sided limits of (f).

Relation to adjacent function classes

Every function of bounded variation on a compact interval is regulated because finite variation forces the existence of one-sided limits. The converse is false. A continuous function may have infinite total variation, and continuity alone already implies regulatedness. For example, the continuous extension of

[ f(x)=x\sin(1/x) ]

at (x=0) is regulated on a compact interval containing zero, while its variation near zero is infinite.

Regulated functions also differ from càdlàg functions. A càdlàg function is right-continuous and has left limits, whereas a regulated function need not agree with either of its one-sided limits at a discontinuity. Every càdlàg function on a compact interval is regulated, but a regulated function becomes càdlàg only after the required right-continuity conditions are imposed.

On a noncompact interval, local regulatedness and global uniform step approximation are distinct. A function may be regulated on every compact subinterval without admitting uniform approximation by a single finite-step structure over the entire domain. This distinction parallels the difference between local boundedness and boundedness in the uniform norm.